![Electrona in crystallo fluentia](https://upload.wikimedia.org/wikipedia/commons/4/4d/Electrona_in_crystallo_fluentia.svg)
The Drude model of
electrical conduction
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
was proposed in 1900 by
Paul Drude
Paul Karl Ludwig Drude (; 12 July 1863 – 5 July 1906) was a German physicist specializing in optics. He wrote a fundamental textbook integrating optics with Maxwell's theories of electromagnetism.
Education
Born into an ethnic German family, D ...
to explain the transport properties of
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s in materials (especially metals). Basically,
Ohm's law was well established and stated that the current ''J'' and voltage ''V'' driving the current are related to the resistance ''R'' of the material. The inverse of the resistance is known as the conductance. When we consider a metal of unit length and unit cross sectional area, the conductance is known as the conductivity, which is the inverse of resistivity. The Drude model attempts to explain the resistivity of a conductor in terms of the scattering of electrons (the carriers of electricity) by the relatively immobile ions in the metal that act like obstructions to the flow of electrons.
The model, which is an application of
kinetic theory, assumes that the microscopic behaviour of electrons in a solid may be treated classically and behaves much like a
pinball
Pinball games are a family of games in which a ball is propelled into a specially designed table where it bounces off various obstacles, scoring points either en route or when it comes to rest. Historically the board was studded with nails call ...
machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.
The two most significant results of the Drude model are an electronic equation of motion,
:
and a linear relationship between
current density and electric field ,
:
Here is the time, ⟨p⟩ is the average momentum per electron and , and are respectively the electron charge, number density, mass, and
mean free time
Molecules in a fluid constantly collide with each other. The mean free time for a molecule in a fluid is the average time between collisions. The mean free path of the molecule is the product of the average speed and the mean free time. These conc ...
between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms why
Ohm's law, one of the most ubiquitous relationships in all of electromagnetism, should hold.
The model was extended in 1905 by
Hendrik Antoon Lorentz
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
(and hence is also known as the Drude–Lorentz model) to give the relation between the
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
and the
electric conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of metals (see
Lorenz number
Lorenz is an originally German name derived from the Roman surname Laurentius, which means "from Laurentum".
Given name
People with the given name Lorenz include:
* Prince Lorenz of Belgium (born 1955), member of the Belgian royal family by hi ...
), and is a
classical model. Later it was supplemented with the results of quantum theory in 1933 by
Arnold Sommerfeld and
Hans Bethe
Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel ...
, leading to the
Drude–Sommerfeld model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with qu ...
.
History
German physicist
Paul Drude
Paul Karl Ludwig Drude (; 12 July 1863 – 5 July 1906) was a German physicist specializing in optics. He wrote a fundamental textbook integrating optics with Maxwell's theories of electromagnetism.
Education
Born into an ethnic German family, D ...
proposed his model in 1900 when it was not clear whether atoms existed, and it was not clear what atoms were on a microscopic scale. The first
direct proof of atoms through the computation of the
Avogadro number
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining co ...
from a microscopic model is due to
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
, the
first modern model of atom structure dates to 1904 and the
Rutherford model
The Rutherford model was devised by the New Zealand-born physicist Ernest Rutherford to describe an atom. Rutherford directed the Geiger–Marsden experiment in 1909, which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum ...
to 1909.
Drude starts from the discovery of electrons in 1897 by
J.J. Thomson
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...
and assumes as a simplistic model of solids that the bulk of the solid is composed of positively charged scattering centers, and a sea of electrons submerge those scattering centers to make the total solid neutral from a charge perspective.
In modern terms this is reflected in the
valence electron
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair form ...
model where the sea of electrons is composed of the valence electrons only, and not the full set of electrons available in the solid, and the scattering centers are the inner shells of tightly bound electrons to the nucleus. The scattering centers had a positive charge equivalent to the
valence number
In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with other atoms when it forms chemical compounds or molecules.
Description
The combining capacity, or affinity of a ...
of the atoms.
This similarity added to some computation errors in the Drude paper, ended up providing a reasonable qualitative theory of solids capable of making good predictions in certain cases and giving completely wrong results in others.
Whenever people tried to give more substance and detail to the nature of the scattering centers, and the mechanics of scattering, and the meaning of the length of scattering, all these attempts ended in failures.
The scattering lengths computed in the Drude model, are of the order of 10 to 100 inter-atomic distances, and also these could not be given proper microscopic explanations.
Drude scattering is not electron-electron scattering which is only a secondary phenomenon in the modern theory, neither nuclear scattering given electrons can be at most be absorbed by nuclei. The model remains a bit mute on the microscopic mechanisms, in modern terms this is what is now called the "primary scattering mechanism" where the underlying phenomenon can be different case per case.
The model gives better predictions for metals, especially in regards to conductivity,
and sometimes is called Drude theory of metals. This is because metals have essentially a better approximation to the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
, i.e. metals do not have complex
band structures
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...
, electrons behave essentially as
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
s and where, in the case of metals, the
effective number of de-localized electrons is essentially the same as the valence number.
The same Drude theory, despite inconsistencies which baffled most physicists of the period, was the major one accepted to explain solids until the introduction in 1927 of the
Drude–Sommerfeld model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with qu ...
.
A few more hints of the correct ingredients of a modern theory of solids was given by the following:
* The
Einstein solid
The Einstein solid is a model of a crystalline solid that contains a large number of independent three-dimensional quantum harmonic oscillators of the same frequency. The independence assumption is relaxed in the Debye model.
While the model provi ...
model and the
Debye model
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
, suggesting that the quantum behaviour of exchanging energy in integral units or
quanta was an essential component in the full theory especially with regard to
specific heats
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
, where the Drude theory failed.
* In some cases, namely in the Hall effect, the theory was making correct predictions if instead of using a negative charge for the electrons a positive one was used. This is now interpreted as holes (i.e. quasi-particles that behave as positive charge carriers) but at the time of Drude it was rather obscure why this was the case.
Drude used
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density ...
for the gas of electrons and for deriving the model, which was the only one available at that time. By replacing the statistics with the correct
Fermi Dirac statistics,
Sommerfeld significantly improved the predictions of the model, although still having a
semi-classical theory that could not predict all results of the modern quantum theory of solids.
Nowadays Drude and
Sommerfeld models are still significant to understanding the qualitative behaviour of solids and to get a first qualitative understanding of a specific experimental setup.
This is a generic method in
solid state physics, where it is typical to incrementally increase the complexity of the models to give more and more accurate predictions. It is less common to use a full-blown
quantum field theory from first principles, given the complexities due to the huge numbers of particles and interactions and the little added value of the extra mathematics involved (considering the incremental gain in numerical precision of the predictions).
Assumptions
Drude used the
kinetic theory of gases applied to the gas of electrons moving on a fixed background of "
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
s"; this is in contrast with the usual way of applying the theory of gases as a neutral diluted gas with no background. The
number density of the electron gas was assumed to be
:
where ''Z'' is the effective number of de-localized electrons per ion, for which Drude used the valence number, ''A'' is the atomic mass per mole,
is the mass density (mass per unit volume)
of the "ions", and ''N'' is the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
.
Considering the average volume available per electron as a sphere:
:
The quantity
is a parameter that describes the electron density and is often of the order of 2 or 3 times the
Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
, for
alkali metals
The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
it ranges from 3 to 6 and some metal compounds it can go up to 10.
The densities are of the order of 100 times of a typical classical gas.
The core assumptions made in the Drude model are the following:
* Drude applied the kinetic theory of a dilute gas, despite the high densities, therefore ignoring electron–electron and electron–ion interactions aside from collisions.
* The Drude model considers the metal to be formed of a collection of positively charged ions from which a number of "free electrons" were detached. These may be thought to be the
valence electrons
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair forms ...
of the atoms that have become delocalized due to the electric field of the other atoms.
* The Drude model neglects long-range interaction between the electron and the ions or between the electrons; this is called the independent electron approximation.
* The electrons move in straight lines between one collision and another; this is called free electron approximation.
* The only interaction of a free electron with its environment was treated as being collisions with the impenetrable ions core.
* The average time between subsequent collisions of such an electron is , with a
memoryless
In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already ...
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
. The nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.
* After a collision event, the distribution of the velocity and direction of an electron is determined by only the local temperature and is independent of the velocity of the electron before the collision event.
The electron is considered to be immediately at equilibrium with the local temperature after a collision.
Removing or improving upon each of these assumptions gives more refined models, that can more accurately describe different solids:
* Improving the hypothesis of the
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density ...
with the
Fermi–Dirac statistics leads to the
Drude–Sommerfeld model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with qu ...
.
* Improving the hypothesis of the Maxwell–Boltzmann statistics with the
Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
leads to considerations about the specific heat of integer spin atoms
and to the
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.6 ...
.
* A valence band electron in a semiconductor is still essentially a free electron in a delimited energy range (i.e. only a "rare" high energy collision that implies a change of band would behave differently); the independent electron approximation is essentially still valid (i.e. no electron–electron scattering), where instead the hypothesis about the localization of the scattering events is dropped (in layman terms the electron is and scatters all over the place).
Mathematical treatment
DC field
The simplest analysis of the Drude model assumes that electric field is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentum between collisions, which occur on average every seconds.
Then an electron isolated at time will on average have been travelling for time since its last collision, and consequently will have accumulated momentum
:
During its last collision, this electron will have been just as likely to have bounced forward as backward, so all prior contributions to the electron's momentum may be ignored, resulting in the expression
:
Substituting the relations
:
:
results in the formulation of Ohm's law mentioned above:
:
Time-varying analysis
![DrudeResponse](https://upload.wikimedia.org/wikipedia/commons/c/cd/DrudeResponse.gif)
The dynamics may also be described by introducing an effective drag force. At time the electron's momentum will be:
:
where
can be interpreted as generic force (e.g.
Lorentz Force) on the carrier or more specifically on the electron.
is the momentum of the carrier with random direction after the collision (i.e. with a momentum
) and with absolute kinetic energy
:
.
On average, a fraction of
of the electrons will not have experienced another collision, the other fraction that had the collision on average will come out in a random direction and will contribute to the total momentum to only a factor
which is of second order.
With a bit of algebra and dropping terms of order
, this results in the generic differential equation
:
The second term is actually an extra drag force or damping term due to the Drude effects.
Constant electric field
At time the average electron's momentum will be
:
and then
:
where denotes average momentum and the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
:
for . The
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
solution, , is then
:
As above, average momentum may be related to average velocity and this in turn may be related to current density,
:
:
and the material can be shown to satisfy Ohm's law
with a
DC-conductivity :
:
AC field
![DrudeModelComplexConductivity](https://upload.wikimedia.org/wikipedia/commons/4/4d/DrudeModelComplexConductivity.png)
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency . The complex conductivity is
:
Here it is assumed that:
:
:
In engineering, is generally replaced by (or ) in all equations, which reflects the phase difference with respect to origin, rather than delay at the observation point traveling in time.
The imaginary part indicates that the current lags behind the electrical field. This happens because the electrons need roughly a time to accelerate in response to a change in the electrical field. Here the Drude model is applied to electrons; it can be applied both to electrons and holes; i.e., positive charge carriers in semiconductors. The curves for are shown in the graph.
If a sinusoidally varying electric field with frequency
is applied to the solid, the negatively charged electrons behave as a plasma that tends to move a distance ''x'' apart from the positively charged background. As a result, the sample is polarized and there will be an excess charge at the opposite surfaces of the sample.
The
dielectric constant of the sample is expressed as
:
where
is the
electric displacement and
is the
polarization density
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
.
The polarization density is written as
:
and the polarization density with ''n'' electron density is
:
After a little algebra the relation between polarization density and electric field can be expressed as
:
The frequency dependent dielectric function of the solid is
:
At a resonance frequency
, called the plasma frequency, the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.
:
The plasma frequency represents a
plasma oscillation Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability ...
resonance or
plasmon
In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...
. The plasma frequency can be employed as a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials.
Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample, a typical example are alkaline metals that becomes transparent in the range of
ultraviolet
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30 PHz) to 400 nm (750 THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
radiation.
Thermal conductivity of metals
One great success of the Drude model is the explanation of the
Wiedemann-Franz law. This was due to a fortuitous cancellation of errors in Drude's original calculation. Drude predicted the value of the Lorenz number:
:
Experimental values are typically in the range of
for metals at temperatures between 0 and 100 degrees Celsius.
Thermopower
A generic temperature gradient when switched on in a thin bar will trigger a current of electrons towards the lower temperature side, given the experiments are done in an open circuit manner this current will accumulate on that side generating an electric field countering the electric current. This field is called thermoelectric field:
:
and Q is called thermopower. The estimates by Drude are a factor of 100 low given the direct dependency with the specific heat.
:
where the typical thermopowers at room temperature are 100 times smaller of the order of micro-volts.
Drude response in real materials
The characteristic behavior of a Drude metal in the time or frequency domain, i.e. exponential relaxation with time constant or the frequency dependence for stated above, is called Drude response. In a conventional, simple, real metal (e.g. sodium, silver, or gold at room temperature) such behavior is not found experimentally, because the characteristic frequency is in the infrared frequency range, where other features that are not considered in the Drude model (such as
band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...
) play an important role.
But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction for . These are materials where the relaxation rate is at much lower frequencies.
This is the case for certain
doped semiconductor single crystals, high-mobility
two-dimensional electron gases, and
heavy-fermion metals.
Accuracy of the model
Historically, the Drude formula was first derived in a limited way, namely by assuming that the charge carriers form a
classical ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
.
Arnold Sommerfeld considered quantum theory and extended the theory to the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
, where the carriers follow
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
. The conductivity predicted is the same as in the Drude model because it does not depend on the form of the electronic speed distribution.
The Drude model provides a very good explanation of DC and AC conductivity in metals, the
Hall effect
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was dis ...
, and the
magnetoresistance
Magnetoresistance is the tendency of a material (often ferromagnetic) to change the value of its electrical resistance in an externally-applied magnetic field. There are a variety of effects that can be called magnetoresistance. Some occur in b ...
in metals near room temperature. The model also explains partly the
Wiedemann–Franz law In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (''κ'') to the electrical conductivity (''σ'') of a metal is proportional to the temperature (''T'').
: \frac \kapp ...
of 1853. However, it greatly overestimates the electronic heat capacities of metals.
In reality, metals and insulators have roughly the same heat capacity at room temperature.
The model can also be applied to positive (hole) charge carriers.
In his original paper, Drude made an error, estimating the
Lorenz number
Lorenz is an originally German name derived from the Roman surname Laurentius, which means "from Laurentum".
Given name
People with the given name Lorenz include:
* Prince Lorenz of Belgium (born 1955), member of the Belgian royal family by hi ...
of Wiedemann–Franz law to be twice what it classically should have been, thus making it seem in agreement with the experimental value of the specific heat. This number is about 100 times smaller than the classical prediction but this factor cancels out with the mean electronic speed that is about 100 times bigger than Drude's calculation.
See also
*
Free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
*
Arnold Sommerfeld
*
Electrical conductivity
Citations
References
General
*
External links
*
{{Atomic models
Electric current
Electric and magnetic fields in matter